Two projectiles A and B are thrown with initial velocities of 40 m s$$^{-1}$$ and 60 m s$$^{-1}$$ at angles 30° and 60° with the horizontal respectively. The ratio of their ranges respectively is $$(g = 10$$ m s$$^{-2}$$)

The range of a projectile is: $$R = \frac{u^2 \sin 2\theta}{g}$$

For projectile A: $$u_A = 40$$ m/s, $$\theta_A = 30°$$

$$ R_A = \frac{40^2 \sin 60°}{10} = \frac{1600 \times \frac{\sqrt{3}}{2}}{10} = 80\sqrt{3}\;\text{m} $$

For projectile B: $$u_B = 60$$ m/s, $$\theta_B = 60°$$

$$ R_B = \frac{60^2 \sin 120°}{10} = \frac{3600 \times \frac{\sqrt{3}}{2}}{10} = 180\sqrt{3}\;\text{m} $$

The ratio of ranges:

$$ \frac{R_A}{R_B} = \frac{80\sqrt{3}}{180\sqrt{3}} = \frac{80}{180} = \frac{4}{9} $$

The correct answer is 4 : 9.